The corresponding predictions of \(e^{\pm }p\)  cross sections for HERA I+II \(e^{\pm }p\)  data [1]} are obtained by solving the DGLAP evolution equations [2]}, [3]}, [4]}, [5]} at next-to-next-to-leading-order (NNLO) in the \(\overline{\mathrm {MS}}\)  scheme [6]} through the QCDNUM program [7]} interfaced with XFITTER. The renormalization and factorization scales are chosen to be \(\mu _r^2\)  = \(\mu _f^2\)  = \(Q^2\) . The initial factorization scale is set to be \(Q_0^2 = 1.9~\mathrm {GeV}^2\)  just below the charm quark mass \(m_c = 1.43~\mathrm {GeV}\)  in this analysis. The evolved PDFs at each scale point are convoluted with the coefficient functions of the structure functions of the proton to obtain the corresponding differential cross sections for \(e^{\pm }p\)  scattering. The contributions to the structure functions from heavy quarks are calculated in the general-mass-variable-flavor-number scheme [8]}, [9]}, [10]} in the NC interactions. For the structure functions in CC interactions, the zero-mass approximation is used because the HERA CC data have \(Q^2\gg M_H^2\) , where \(M_H\)  is the heavy quark mass, i.e., the masses of charm and bottom quarks. The calculation of the differential cross section for the gauge boson production is done using the well-known Monte Carlo program MCFM [11]}, [12]} with the CT18NNLO PDF set [13]}. The \(W^{\pm }\)  and \(Z\)  boson cross sections are available up to NNLO. Because of the intensiveness of this calculation, a fast convolution technique is applied. First of all, the partonic cross sections in \(pp\)  collision are obtained in the form of grid files using MCFM with the help of another fast parton convolution program APPLGRID[14]} interfaced both with MCFM and XFITTER; then, the corresponding cross sections are obtained convoluting the grid files with the evolved PDFs during the fit. So far, APPLGRID (version-1.5.34) can only calculate cross sections up to next-to-leading-order (NLO) accuracy. To get the NNLO total and differential cross sections, a \(K\) -factor formalism is applied. The \(K\) -factor is defined as
\(K= \frac{\sigma ^{\mathrm {NLO~EW}}_{\mathrm {NNLO~QCD}}}{\sigma ^{\mathrm {LO~EW}}_{\mathrm {NLO~QCD}}}\) 
